Comparing methods for calculating Z-factor

Comparing methods for calculating Z-factor Gabor Takacs

nal points for pseudoreduced temperature (Tr) and pseudoreduced pressure (Pr) in the ranges of 1.2 -:; T, -:; 3.0 and 0.2 -:; p, -:; 15.0. Petroleum engineering calculations often require knowledge of Z-factors for natural gases, but experimental data from pressure-volume-temperature (p - V - T) measurements are seldom available. In such cases, use of the Standing-Katz Z-factor chart' or its tabulated form is generally accepted. This chart gives reasonably accurate results for mixtures composed of pure hydrocarbon gases. The use of this chart in complex calculation schemes is often too timeconsuming or perhaps impossible.

Texas Tech University Lubbock, Tex. From an examination of computational methods, one can select the most accurate method for calculating a gas deviation factor (Z), the method that needs the least computer time, and whether a minicomputer or larger machine is needed. This examination covered thirteen computational methods for describing the Standing-Katz natural gas deviation factor chart that has been used for more than 40 years. The accuracy of the methods has been determined based on 180 origi-

Thus the need arose early for some kind of mathematical description of that chart. Furthermore, numerical methods are of utmost importance for computer application of any calculation involving determination of Z-factors.

Methods investigated The most commonly used methods to describe the original Z-factor chart are evaluated in this article which is an extension of a study published in a previous article by the author." Following is a review of the methods investigated: Gray-Sirns.:' This method involves the storage of a matrix composed of Z Table 1

Comparison of methods for computer calculation of Z-factors 1. Gray-Sims (1959)

10. Oranchuk-A. Kassem (1975)

5. Hankinson, et al. (1969) 6. Carlile-Gillett (1971)

2. Sarem (1969)

11. Gopal (1977)

7. Hall-Yarborough (1973) 3. Leung (1964) 4. Papay (1968)

3

Average error, %

0.145

-0.043

0.638

Average absolute error,%

0.190

0.939

2.115

0.718

1.08 a

0.656

4 -4.889

6

11

7

8

9

10

-0.158

-3.423

-0.017

-0.002

0.105

12 -3.882

13 0.120

-0.052

7 .• 69

2.799

0.208

0_ 512

3.966

0.361

0.304

1.338

4.601

0.539

0.362

1. 961

0.329

1.798

0.659

1. 588

1. 231

0.294

0.333

0.367

0.417 0.402 0.361 0.169 0.313 0.424 0.293 0.281 0.264 0.665

0.335 0.417 0.308 0.143 0.244 0.302 0.254 O. J83 0.221 0.633

1.177 1. 251 0.745 1.008 1.16J 1. 497 1.588 2.210 0.461 2.279

20.639 2.251 2.070 1. 441 1. 3 79 1.204 1. 215 2.J 56 4.920 8.740

1. 068 0.547 0.790 0.877 0.45J 0.350 0.325 0.310 0.287 0.382

0.088 0.151 0.299 0.424 0.483 0.28S 0.337 0.514 0.701 0.665 0.569 0.434 0.242 0.125 0.155 0.202 0.247 0.575

0.079 0.165 0.265 0.323 0.369 0.234 0.317 0.424 0.574 0.528 0.477 0.372 0.247 0.182 0.146 0.185 0.159 0.424

0.121 0.190 0.514 0.775 1. 238 0.730 1.656 0.714 0.899 0.725 0.684 3.590 3.008 2.522 2.098 1. 602 1.301 1. 710

Isotherm average absolute error, % 0.070 0.081 0.075 0.048 0.063 0.437 0.435 0.354 0.324 0.016

2.160 1.370 0.991 0.787 0.742 0.6ld 0.561 0.43. 0.654 1.071

0.055 0.050 0.047 0.015 0.031 0.000 0.046 O.ll64 0.014 0.049 0.039 0.255 0.495 u.678 0.854 0.700 0.035 0.000

1.147 0.529 1.176 0.880 0.957 1.567 1. 206 0.738 0.696 0.622 0.861 1. 054 loll 7 0.934 0.586 0.217 0.756 1. ass

7.689 4.179 2.558 1.704 1.183 0.655 0.479 0.472 0.532 1. 500

19.490 12.541 9.518 8.564 8.023 7.067 5.702 3.107 1. 209 4.171

11.387 6.176 3.497 2.193 1.469 1. 093 0.802 o . 553 0.299 0.524

0.235 0.389 562 1. J72 2. GdS 3.1159 2.132 2.021 3. Old J. e76 4. :01 4. i76 4. SOB 4.695 5.1d5 6.968 20.450 72.409

0.181 0.425 0.782 0.987 0.564 1.199 1. 836 2. J64 2.903 1.179 3.598 2.078 2.429 2.876 3.346 4.443 6.395 10.80 0

0.334 0.214 0.179 0.115 0.259 0.216 0.182 0.233 0.191 0.153

0.552 0.620 0.425 0.602 0.424 0.341 0.381 0.489 0.682 0.607

1. 627 1. 042

o . 372 0.857 0.724 0.425 0.447 1.909 7.149 25.J04

Isobar average absolute error

P, 0.2 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7. a 8.0 10.0 15.0

5 2.261

T, 1.2 1.3 1.4 1.5 1.6 1.7 1.6 2.0 2.4 3.0

13. Papp (1979)

9. Oranchuk, el al. (1974)

2

Average running time, sec/Z value

12. Burnett (1979)

8. Brill (1974)

1. 689 1.1&5 0.825 0.577 1. sos 2. 07d 1. 502

O.BoO 1.078 1. 091 1. 046 0.867 0.638 0.330 0.413 1. 50b 2.275 18.480

a.

0.129 0.118 0.065 0.138 0.160 0.149 0.200 0.161 0.134 0.115 0.127 0.098 0.11 0 0.087 0.251 0.659 0.708 0.330

1. 301 1. 553 0.567 0.422 0.583 0.577 0.575 0.447 0.437 0.331 0.325 0.310 0.503 0.304 0.242 0.258 0.109 0.381

0.292 0.591 1.175 2.014 3.136 3.635 3.445 3.344 3.547 3.757 3.983 4.249 4.359 4.495 4.636 4.998 6.100 13.622

0.627 1. 488 2.842 3.844 J. 66 9 4.303 6.680 8.359 9.603

0.230 0.375 0.268 0.387 0.89J 0.9J8 0.703 0.658 0.650 0.532 e.465 0.490 0.579 0.6J3 0.631 0.616 0.319 0;·367

OGJ

TECHNOLOGY

May 15, 1989, Oil & Gas Journal 43

values taken from the Standing-Katz chart at some fixed values of pseudoreduced parameters p, and T,. An interpolation scheme is then used to compute deviation factors from the stored values. To minimize computer memory requirements, data points were taken from the original chart by taking into account its varying curvature. Thus, a 20 x 20 matrix of Z values was constructed that adequately represents the whole chart. This method requires the storage in memory of one 20 x 20-type matrix along with two vectors of 20 elements each. Therefore, the use of most types of programmable calculators is heavily restricted, as their storage capacity is relatively small. The ranges of applicability of this Zfactor calculation method are the following, according to the authors:

Table 2

Three different

1121REM SUBROUTINE TO CALCULATE 20 REM using the method of 30 REM 40 Z0"".3379·I>LOG(LOG

40 DATA 1.6643,-2.2114,-0.3647,1.4385.0.5222,-0.8511,-0. 0364. 1.049 50 DATA a. 1391.-0.2988,0.0007,0. 9969,0.11J295. -0. 0825,0. 121009,0. 9967 60 OAT A -1.357,1.4942,4.6315,-4.7009,0.1717,-0.3232,0.5869,0.1229 70 DATA 0.0984,-0.2053,0.062190.858,0.0211,-0.0527,0.0127 ,0. 9549 80 DATA -0.3278,O. 47~2,1. 8223,-1. 9036,-0. 2521,0. 3871,1.6087,-1.6635 90 DArA -0.0284 ,0. 062~,0. 4714. -0. 0011 ,0.0041,0.0039,0.0607,0.7927 100 FOR 19=1 TO 48 110 READ A(I9) 12121NEXT 19 130 IF PRED(=5. -4 THEN 16Q1 140 Z""PRED*{ . 711 +3. 66*TRED)·" (-1. 150 RETURN 160 IF PRED2.8 THEN 230 180 IF PRED >1.4 THEN 250 190IFTRED> 1.19HtEN250 200 IF TRED 1 . 08 THEN 2~0 210 19=1 220 GOTO 260 230 19:3

4667) -1.637/

(. 319*TRED· +, 522)

+2. 071